Search results

This paper seeks to develop, propose and validate, through a series of presentable examples, a comprehensive high‐precision and ultra‐fast computing concept for solving stiff ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN).

The core of the concept developed in this paper is a straight‐forward scheme that we call “nonlinear adaptive optimization (NAOP)”, which is used for a precise template calculation for solving any (stiff) nonlinear ODEs through CNN processors.

One of the key contributions of this work (this is a real breakthrough) is to demonstrate the possibility of mapping/transforming different types of nonlinearities displayed by various classical and well‐known oscillators (e.g. van der Pol‐, Rayleigh‐, Duffing‐, Rössler‐, Lorenz‐, and Jerk‐ oscillators, just to name a few) unto first‐order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN‐templates. Furthermore, in case of PDEs solving, the same concept also allows a mapping unto first‐order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDEs in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultra‐fast solver of stiff differential equations (both ODEs and PDEs). This clearly enables a CNN‐based, real‐time, ultra‐precise, and low‐cost Computational Engineering. As proof of concept a well‐known prototype of stiff equations (van der Pol) has been considered; the corresponding precise CNN‐templates are derived to obtain precise solutions of this equation.

This paper contributes to the enrichment of the literature as the relevant state‐of‐the‐art does not provide a systematic and robust method to solve nonlinear ODEs and/or nonlinear PDEs using the CNN‐paradigm. Further, the “NAOP” concept developed in this paper has been proven to perform accurate and robust calculations. This concept is not based on trial‐and‐error processes as it is the case for various classes of optimization methods/tools (e.g. genetic algorithm, particle swarm, neural networks, etc.). The “NAOP” concept developed in this frame does significantly contribute to the consolidation of CNN as a universal and ultra‐fast solver of nonlinear differential equations (both ODEs and PDEs). An implantation of the concept developed is possible even on embedded digital platforms (e.g. field‐programmable gate array (FPGA), digital signal processing (DSP), graphics processing unit (GPU), etc.); this opens a broad range of applications. On‐going works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting PDE models such as Navier Stokes, Schrödinger, Maxwell, etc.

Ordinary differential equations (ODEs) are widely used in the engineering curriculum. They model a spectrum of interesting physical problems that arise in engineering disciplines. Studies of different types of ODEs are determined by engineering applications. Various techniques are used to solve practical differential equations problems. This paper aims to present a computational tool or a computer-assisted technique aimed at tackling ODEs. This method is usually not taught and/or not accessible to undergraduate students. The aim of this strategy is to help the readers to develop an effective and relatively novel problem-solving skill. Because of the drudgery of hand computations involved, the method requires the need to use computers packages. In this work, the successive differentiation method (SDM) for solving linear and nonlinear and homogeneous or non-homogeneous ODEs is presented. The algorithm uses the successive differentiation of any given ODE to determine the values of the function’s derivatives at a single point, mostly x = 0. The obtained values are used to construct the Taylor series of the solution of the examined ODE. The algorithm does not require any new assumption, hence handles the problem in a direct manner. The power of the method is emphasized by testing a variety of models with distinct orders, with constant and variable coefficients. Most of the symbolic and numerical computations can be carried out using computer algebra systems.

This study presents a computational tool or a computer-assisted technique aimed at tackling ODEs. This method is usually not taught and/or not accessible to undergraduate students. The aim of this strategy is to help the readers to develop an effective and relatively novel problem-solving skill. Because of the drudgery of hand computations involved, the method requires the need to use computers packages.

This method is applied to a variety of well-known equations, such as the Bernoulli equation, the Riccati equation, the Abel equation and the second-order Euler equation, some with constant and variable coefficients. SDM handles linear and nonlinear and homogeneous or nonhomogeneous ODEs in a direct manner without any need to restrictive conditions. The method works effectively to the Volterra integral equations, as will be discussed in a coming work.

The method can be extended to a wide range of engineering problems that are modeled by differential equations. The method is simple and novel and highly accurate.

The purpose of this paper is to analyse a novel technique for an efficient numerical approximation of systems of highly oscillatory ordinary differential equations (ODEs) that arise in electronic systems subject to modulated signals.

The paper combines a Filon‐type method with waveform relaxation techniques for nonlinear systems of ODEs.

The analysis includes numerical examples to compare with traditional methods such as the trapezoidal rule and Runge‐Kutta methods. This comparison shows that the proposed approach can be very effective when dealing with systems of highly oscillatory differential equations.

The present paper constitutes a preliminary study of Filon‐type methods applied to highly oscillatory ODEs in the context of electronic systems, and it is a starting point for future research that will address more general cases.

The proposed method makes use of novel and recent techniques in the area of highly oscillatory problems, and it proves to be particularly useful in cases where standard methods become expensive to implement.

The purpose of this study is to analyze the effects of carbon nanotubes (CNTs) in the Marangoni convection boundary layer viscous fluid flow. The analysis and formulation for both types of CNTs, namely, single-walled (SWCNTs) and multi-walled (MWCNTs), are described. The influence of thermal radiation effect assumed in the form of energy expression.

Appropriate transformations reduced the partial differential systems to a set of nonlinear ordinary differential equations (ODEs). The obtained nonlinear ODE set is solved via the least squares method. A detailed comparison between outcomes obtained by the least squares method, RK-4 and already published work is available.

Nusselt number was analyzed and found to be more effective for nanoparticle volume fraction and larger radiation parameters. Additionally, the error and convergence analysis for the least squares method was presented to show the efficiency of the said algorithm.

The results reveal that velocity is a decreasing function of suction for both CNTs. While enhancing the nanoparticle volume fraction, an increase for both thermal boundary layer thickness and temperature was attained. The radiation parameter has an increasing function as temperature. Velocity behavior is the same for nanoparticle volume fraction and suction. It was observed that velocity is less in SWCNTs as compared to MWCNTs.

In this study, a second-order standard wave equation extended to a two-dimensional viscous wave equation with timely differentiated advection-diffusion terms has been solved by differential quadrature methods (DQM) using a modification of cubic B-spline functions. Two numerical schemes are proposed and compared to achieve numerical approximations for the solutions of nonlinear viscous wave equations.

Two schemes are adopted to reduce the given system into two systems of nonlinear first-order partial differential equations (PDE). For each scheme, modified cubic B-spline (MCB)-DQM is used for calculating the spatial variables and their derivatives that reduces the system of PDEs into a system of nonlinear ODEs. The solutions of these systems of ODEs are determined by SSP-RK43 scheme. The CPU time is also calculated and compared. Matrix stability analysis has been performed for each scheme and both are found to be unconditionally stable. The results of both schemes have been extensively discussed and compared. The accuracy and reliability of the methods have been successfully tested on several examples.

A comparative study has been carried out for two different schemes. Results from both schemes are also compared with analytical solutions and the results available in literature. Experiments show that MCB-DQM with Scheme II yield more accurate and reliable results in solving viscous wave equations. But Scheme I is comparatively less expensive in terms of CPU time. For MCB-DQM, less depository requirements lead to less aggregation of approximation errors which in turn enhances the correctness and readiness of the numerical techniques. Approximate solutions to the two-dimensional nonlinear viscous wave equation have been found without linearizing the equation. Ease of implementation and low computation cost are the strengths of the method.

For the first time, a comparative study has been carried out for the solution of nonlinear viscous wave equation. Comparisons are done in terms of accuracy and CPU time. It is concluded that Scheme II is more suitable.

The purpose of this paper is to use the polynomial differential quadrature method (PDQM) to find the numerical solutions of some Burgers'‐type nonlinear partial differential equations.

The PDQM changed the nonlinear partial differential equations into a system of nonlinear ordinary differential equations (ODEs). The obtained system of ODEs is solved by Runge‐Kutta fourth order method.

Numerical results for the nonlinear evolution equations such as 1D Burgers', coupled Burgers', 2D Burgers' and system of 2D Burgers' equations are obtained by applying PDQM. The numerical results are found to be in good agreement with the exact solutions.

A comparison is made with those which are already available in the literature and the present numerical schemes are found give better solutions. The strong point of these schemes is that they are easy to apply, even in two‐dimensional nonlinear problems.

The simplified modeling of many physical processes results in a second-order ordinary differential equation (ODE) system. Often the damping of these resonating systems cannot be defined in the same simplified way as the other parameters due to the complexity of the physical effects. The purpose of this paper is to develop a mathematically stable approach for damping resonances in nonlinear ODE systems.

Modifying the original ODE using the eigenvalues and eigenvectors of a linearized state leads to satisfying results.

An iterative approach is presented, how to modify the original ODE, to achieve a well-damped solution.

The method can be applied for every physical resonating system, where the model complexity prevents the determination of the damping.

The iterative algorithm to modify the original ODE is novel. It can be used on different fields of the physics, where a second-order ODE is describing the problem, which has only measured or empirical damping.

In this communication, a theoretical simulation is aimed to characterize the Darcy–Forchheimer flow of a magneto-couple stress fluid over an inclined exponentially stretching sheet. Stokes’ couple stress model is deployed to simulate non-Newtonian microstructural characteristics. Two different kinds of thermal boundary conditions, namely, the prescribed exponential order surface temperature (PEST) and prescribed exponential order heat flux, are considered in the heat transfer analysis. Joule heating (Ohmic dissipation), viscous dissipation and heat source/sink impacts are also included in the energy equation because these phenomena arise frequently in magnetic materials processing.

The governing partial differential equations are transformed into nonlinear ordinary differential equations (ODEs) by adopting suitable similar transformations. The resulting system of nonlinear ODEs is tackled numerically by using the Runge–Kutta fourth (RK4)-order numerical integration scheme based on the shooting technique. The impacts of sundry parameters on stream function, velocity and temperature profiles are viewed with the help of graphical illustrations. For engineering interests, the physical implication of the said parameters on skin friction coefficient, Nussult number and surface temperature are discussed numerically through tables.

As a key outcome, it is noted that the augmented Chandrasekhar number, porosity parameter and Forchhemeir parameter diminish the stream function as well as the velocity profile. The behavior of the Darcian drag force is similar to the magnetic field on fluid flow. Temperature profiles are generally upsurged with the greater magnetic field, couple stress parameter and porosity parameter, and are consistently higher for the PEST case.

The findings obtained from this analysis can be applied in magnetic material processing, metallurgy, casting, filtration of liquid metals, gas-cleaning filtration, cooling of metallic sheets, petroleum industries, geothermal operations, boundary layer resistors in aerodynamics, etc.

From the literature review, it has been found that the Darcy–Forchheimer flow of a magneto-couple stress fluid over an inclined exponentially stretching surface with heat flux conditions is still scarce. The numerical data of the present results are validated with the already existing studies under limited cases and inferred to have good concord.

This paper aims to deal with two-dimensional magneto-hydrodynamic (MHD) Falkner–Skan boundary layer flow of an incompressible viscous electrically conducting fluid over a permeable wall in the presence of a magnetic field.

Using the Lie group approach, the Lie algebra of infinitesimal generators of equivalence transformations is constructed for the equation under consideration. Using these suitable similarity transformations, the governing partial differential equations are reduced to linear and nonlinear ordinary differential equations (ODEs). Further, Haar wavelet approach is applied to the reduced ODE under the subalgebra 4.1 for constructing numerical solutions of the flow problem.

A new type of solutions was obtained of the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis.

To find a solution for the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis is a new approach for fluid problems.

The purpose of this paper is to investigate a three-dimensional boundary layer flow with considering heat and mass transfer on a nonlinearly stretching sheet by using a novel operational-matrix-based method.

The partial differential equations that governing the problem are converted into the system of nonlinear ordinary differential equations (ODEs) with considering suitable similarity transformations. A direct numerical method based on the operational matrices of integration and product for the linear barycentric rational basic functions is used to solve the nonlinear system of ODEs.

Graphical and tabular results are provided to illustrate the effect of various parameters involved in the problem on the velocity profiles, temperature distribution, nanoparticle volume fraction, Nusselt and Sherwood number and skin friction coefficient. Comparison between the obtained results, numerical results based on the Maple's dsolve (type = numeric) command and previous existing results affirms the efficiency and accuracy of the proposed method.

The motivation of the present study is to provide an effective computational method based on the operational matrices of the barycentric cardinal functions for solving the problem of three-dimensional nanofluid flow with heat and mass transfer. The convergence analysis of the presented scheme is discussed. The benefit of the proposed method (PM) is that, without using any collocation points, the governing equations are converted to the system of algebraic equations.